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Length Contraction

As the velocity of an object approaches the speed of light it appears to change length to a stationary observer.

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Length Contraction

Students will learn how length contraction follows from time dilation and the meaning of the gamma factor.

Key Equations

\begin{align*} \beta = \frac{v}{c}\end{align*}β=vc

An object moving with speed \begin{align*}v\end{align*}v has a dimensionless speed \begin{align*} \beta \end{align*}β calculated by dividing the speed \begin{align*}v\end{align*}v by the speed of light (\begin{align*}\mathit{c} = 3\times10^8 \;\mathrm{m/s}\end{align*}). \begin{align*}0 \le \beta \le 1\end{align*}.

\begin{align*} \gamma = \frac{1}{\sqrt{1-\beta^2}}\end{align*}

The dimensionless Lorentz “gamma” factor \begin{align*}\gamma\end{align*} can be calculated from the speed, and tells you how much time dilation or length contraction there is. \begin{align*}1 \le \gamma \le \infty \end{align*}.

\begin{align*}L' = \frac{L}{\gamma} \end{align*}

If you see an object of length\begin{align*}L\end{align*} moving towards you at a Lorentz gamma factor \begin{align*}\gamma\end{align*}, it will appear shortened (contracted) in the direction of motion to new length \begin{align*}L\end{align*} .

Clocks moving towards or away from you run more slowly, and objects moving towards or away from you shrink in length. These are known as Lorentz time dilation and length contraction; both are real, measured properties of the universe we live in.
Object Speed (km/sec) \begin{align*}\beta\end{align*} Lorentz \begin{align*}\gamma\end{align*} Factor
Commercial Airplane \begin{align*}0.25\end{align*} \begin{align*}8\times10^{-7}\end{align*} \begin{align*}1.00000000000\end{align*}
Space Shuttle \begin{align*}7.8\end{align*} \begin{align*}3\times10^{-5}\end{align*} \begin{align*}1.00000000034\end{align*}
UFO ☺ \begin{align*}150,000\end{align*} \begin{align*}0.5\end{align*} \begin{align*}1.15\end{align*}
Electron at the Stanford Linear Accelerator \begin{align*}\sim300,000\end{align*} \begin{align*}0.9999999995\end{align*} \begin{align*}\sim100,000\end{align*}

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Time for Practice

  1. What would be the Lorentz gamma factor \begin{align*}\gamma\end{align*} for a space ship traveling at the speed of light c? If you were in this space ship, how wide would the universe look to you?
  2. How fast would you have to drive in your car in order to make the road \begin{align*}50\end{align*}% shorter through Lorentz contraction?
  3. In 1987 light reached our telescopes from a supernova that occurred in a near-by galaxy \begin{align*}160,000\end{align*} light years away. A huge burst of neutrinos preceded the light emission and reached Earth almost two hours ahead of the light. It was calculated that the neutrinos in that journey lost only \begin{align*}13\end{align*} minutes of their lead time over the light.
    1. What was the ratio of the speed of the neutrinos to that of light?
    2. Calculate how much space was Lorentz-contracted form the point of view of the neutrino.
    3. Suppose you could travel in a spaceship at that speed to that galaxy and back. It that were to occur the Earth would be \begin{align*}320,000\end{align*} years older. How much would you have aged?

Answers to Selected Problems

  1. \begin{align*}\gamma = \infty\end{align*}, the universe would be a dot
  2. \begin{align*}9.15 \times 10^7\;\mathrm{m/s}\end{align*}
  3. .

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